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On complete moment convergence for nonstationary negatively associated random variables
- Mi-Hwa Ko1Email author
Journal of Inequalities and Applications20162016:131
https://doi.org/10.1186/s13660-016-1074-4
© Ko 2016
Received: 28 August 2015
Accepted: 26 April 2016
Published: 4 May 2016
Abstract
The purpose of this paper is to establish the complete moment convergence for nonstationary negatively associated random variables satisfying the weak mean domination condition. The result is an improvement of complete convergence in Marcinkiewicz-Zygmund-type SLLN for negatively associated random variables in Kuczmaszewska (Acta Math. Hung. 128:116-130, 2010).
Keywords
- complete moment convergence
- negative association
- nonstationary
- weak mean domination
MSC
- 60F15
1 Introduction
A sequence of random variables \(\{X_{n}, n\geq1\}\) is said to converge completely to a constant c if \(\sum_{n=1}^{\infty}P(|X_{n}-c|>\epsilon )<\infty\) for all \(\epsilon>0\). This concept of complete convergence was introduced by Hsu and Robbins [2]. They proved that the sequence of arithmetic means of independent identically distributed random variables converges completely to the expected value of the summands, provided that the variance is finite. The converse theorem was proved by Erdös [3]. This Hsu-Robbins-Erdös’s result was generalized in different ways. Katz [4], Baum and Katz [5], and Chow [6] obtained a generalization of complete convergence for a sequence of independent identically distributed random variables with normalization of Marcinkiewicz-Zygmund type (see Gut [7]). Chow [8] first showed the complete moment convergence for a sequence of i.i.d. random variables by generalizing the result of Baum and Katz [5].
The concept of complete moment convergence is as follows. Let \(\{Y_{n}, n\geq1\}\) be a sequence of random variables, and \(a_{n}>0\), \(b_{n}>0\), \(q>0\).
If \(\sum_{n=1}^{\infty}a_{n} E\{b_{n}^{-1}|Y_{n}|-\epsilon\}_{+}^{q}<\infty\) for all \(\epsilon>0\), then \(\{Y_{n}, n\geq1\}\) is called complete moment convergence. It is well known that the complete moment convergence implies the complete convergence.
Since then, many investigations have been made. For example, Sung [9] obtained a moment inequality for the maximum partial sum of random variables and the complete moment convergence of i.i.d. random variables, Liang and Li [10] provided necessary and sufficient moment conditions for the complete moment convergence of identically distributed negatively associated random variables, and Wu et al. [11] established the complete moment convergence of identically distributed \(\rho^{*}\)-mixing random variables.
The concept of negative association was introduced by Joag-Dev and Proschan [12] in the following way. A finite family of random variables \(\{X_{i}, 1\leq i \leq n\}\) is said to be negatively associated if for every pair of disjoint nonempty subsets A and B of \(\{1,2,\ldots, n\}\) and any real coordinatewise nondecreasing functions f and g, whenever f and g are such that the covariance exists. An infinite family of random variables is negatively associated if every finite subfamily is negatively associated.
$$ \operatorname{Cov}\bigl(f(X_{i}, i\in A), g(X_{j}, j\in B)\bigr)\leq0 $$
(1.1)
A sequence of random variables \(\{X_{n}, n\geq1\}\) is said to satisfy a weak mean dominating condition with mean dominating random variable X if for some \(c>0\),
$$ \frac{1}{n}\sum_{i=1}^{n} P\bigl(|X_{i}|>x\bigr)\leq cP\bigl(|X|>x\bigr) \quad \mbox{for all } x>0 \mbox{ and all } n\geq1. $$
(1.2)
Recently, Kuczmaszewska [1] proved the complete convergence for a sequence of negatively associated random variables satisfying (1.2) (see Lemma 2.3).
In this paper, we prove the complete moment convergence for sequences of negatively associated random variables satisfying a weak mean dominating condition. It provides an improvement of complete convergence for negatively associated random variables by Kuczmaszewska [1] and conditions for complete moment convergence for a nonstationary sequence of negatively associated random variables.
2 Preliminaries
In our further considerations, we need the following lemmas.
Lemma 2.1
(Shao [13])
Let
\(\{X_{i}, 1\leq i \leq n\}\)
be a sequence of negatively associated random variables with
\(EX_{i}=0\)
and
\(E|X_{i}|^{q}<\infty\)
for
\(q>1\)
and every
\(1\leq i\leq n\). Then
and
$$ E\max_{1\leq k\leq n}\Biggl|\sum_{i=1}^{k} X_{i}\Biggr|^{q}\leq2^{3-q}\sum _{i=1}^{n} E|X_{i}|^{q} \quad \textit{for } 1< q\leq2 $$
(2.1)
$$ E\max_{1\leq k\leq n}\Biggl|\sum_{i=1}^{k} X_{i}\Biggr|^{q}\leq2\biggl(\frac{15 q}{\ln q}\biggr)^{q} \Biggl\{ \sum_{i=1}^{n} E|X_{i}|^{q}+ \Biggl(\sum_{i=1}^{n}EX_{i}^{2} \Biggr)^{\frac{q}{2}}\Biggr\} \quad\textit{for } q> 2. $$
(2.2)
Remark
In Lemma 2.1 for \(q=1\), we have
$$E\max_{1\leq k\leq n}\Biggl|\sum_{i=1}^{k} X_{i}\Biggr|\leq\sum_{i=1}^{n} E|X_{i}|. $$
Lemma 2.2
(Gut [7])
Let \(\{X_{n}, n\geq1\}\) be a sequence of random variables satisfying a weak mean dominating condition with mean dominating random variable X, that is, for some \(c>0\), (1.2) holds for all \(x>0\) and \(n\geq1\).
Let
\(r>0\)
and, for some
\(A>0\),
and
Then, for some constant
\(C>0\),
$$\begin{aligned}& X_{i}^{\prime}=X_{i} I\bigl(|X_{i}|\leq A\bigr), \qquad X_{i}^{\prime\prime}=X_{i} I\bigl(|X_{i}|>A\bigr), \\& X_{i}^{*}=X_{i} I\bigl(|X_{i}|\leq A\bigr)-AI(X_{i}< -A)+AI(X_{i}>A) \end{aligned}$$
$$\begin{aligned}& X^{\prime}=X I\bigl(|X|\leq A\bigr), \qquad X^{\prime\prime}=X I\bigl(|X|>A\bigr), \\& X^{*}=X I\bigl(|X|\leq A\bigr)-AI(X< -A)+AI(X>A) . \end{aligned}$$
- (i)
if \(E|X|^{r}<\infty\), then \((n^{-1})\sum_{i=1}^{n} E|X_{i}|^{r}\leq CE|X|^{r}\),
- (ii)
\((n^{-1})\sum_{i=1}^{n} E|X_{i}^{\prime}|^{r}\leq C(E|X^{\prime}|^{r}+A^{r} P(|X|>A))\) for all \(A>0\),
- (iii)
\((n^{-1})\sum_{i=1}^{n} E|X_{i}^{\prime\prime}|^{r}\leq CE|X^{\prime\prime}|^{r}\) for all \(A>0\),
- (iv)
\((n^{-1})\sum_{i=1}^{n} E|X_{i}^{*}|^{r}\leq CE|X^{*}|^{r}\) for all \(A>0\).
The following result is obtained by Theorem 2.1 of Kuczmaszewska [1].
Lemma 2.3
(Kuczmaszewska [1])
Let
\(\alpha p>1\), \(p>0\), and
\(\alpha>\frac{1}{2}\). Let
\(\{X_{n}, n\geq1\}\)
be a sequence of negatively associated random variables with
\(EX_{n}=0\)
for all
\(n\geq 1\), and
X
be a random variable possibly defined on a different space satisfying condition (1.2) for all
\(\epsilon>0\)
and
\(n\geq1\). Then
implies
$$ E|X|^{p}< \infty $$
(2.3)
$$ \sum_{n=1}^{\infty}n^{\alpha p-2}P\Biggl( \max_{1\leq j\leq n}\Biggl|\sum_{i=1}^{j} X_{i}\Biggr|>\epsilon n^{\alpha}\Biggr)< \infty \quad\textit{for all } \epsilon>0. $$
(2.4)
Lemma 2.4
Let
\(\alpha p>1\), \(\alpha>\frac{1}{2}\), and
\(p>0\). Let
\(\{X_{n}, n\geq1\}\)
be a sequence of negatively associated random variables with
\(EX_{n}=0\)
for all
\(n\geq1\), and
X
be a random variable possibly defined on a different space satisfying condition (1.2) for all
\(\epsilon>0\)
and
\(n\geq1\). Then (2.3) implies
$$ \sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}P\Biggl(\max_{1\leq k\leq n}\Biggl| \sum_{i=1}^{k} X_{i}\Biggr|>u\Biggr)\,du< \infty. $$
(2.5)
Proof
In the case \(0< p<1\), let, for all \(u>0\), and Then we obtain
$$X_{i}=X_{i}I\bigl[|X_{i}|\leq u\bigr]+X_{i}I\bigl[|X_{i}|> u\bigr]=X_{ui}+X_{ui}^{\prime} $$
$$S_{n}=\sum_{i=1}^{n} X_{i}I\bigl[|X_{i}|\leq u\bigr]+\sum_{i=1}^{n} X_{i}I\bigl[|X_{i}|>u\bigr]=\sum_{i=1}^{n} X_{ui}+\sum_{i=1}^{n} X_{ui}^{\prime}. $$
$$\begin{aligned} &\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}P\Bigl(\max_{1\leq j\leq n}|S_{j}|>u \Bigr)\,du \\ &\quad\leq\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}P\Biggl(\max_{1\leq j\leq n}\Biggl| \sum_{i=1}^{j} X_{ui}\Biggr|> \frac{u}{2}\Biggr)\,du \\ &\qquad{} +\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}P\Biggl(\max_{1\leq j\leq n}\Biggl| \sum_{i=1}^{j} X_{ui}^{\prime}\Biggr|> \frac{u}{2}\Biggr)\,du \\ &\quad:=I+J. \end{aligned}$$
(2.6)
For J, take \(q=p\) (\(0< p<1\)). By the Markov inequality, Lemma 2.2(iii), and \(E|X|^{p}<\infty\) we have that Similarly, for I, take \(q=p\) (\(0< p<1\)). By the Markov inequality and Lemma 2.2(ii) we have that In the processing of (2.7), we obtain \(I_{2}<\infty\). It remains to prove that \(I_{1}<\infty\): Hence, from (2.6)-(2.8) the result (2.5) follows in the case \(0< p<1\).
$$\begin{aligned} J \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-\frac{q}{2}}E\Biggl(\max _{1\leq j\leq n}\Biggl|\sum_{i=1}^{j} X_{ui}^{\prime}\Biggr|\Biggr)^{\frac {q}{2}}\,du \quad \mbox{by Markov inequality} \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-\frac{q}{2}}\sum _{i=1}^{n} E\bigl|X_{ui}^{\prime}\bigr|^{\frac{q}{2}}\,du\quad \biggl(\frac {q}{2}< \frac{1}{2}\biggr) \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-\frac{q}{2}}\sum _{i=1}^{n} E|X_{i}|^{\frac{q}{2}}I\bigl(|X_{i}|>u\bigr)\,du \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha} \int_{n^{\alpha}}^{\infty}u^{-\frac{q}{2}}E|X|^{\frac{q}{2}}I\bigl(|X|>u\bigr)\,du \quad \mbox{by Lemma}~2.2(\mathrm{iii}) \\ =&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha}\sum _{m=n}^{\infty}\int _{m^{\alpha}}^{(m+1)^{\alpha}} u^{-\frac{p}{2}}E|X|^{\frac {p}{2}}I\bigl(|X|>u\bigr)\,du\quad (q=p) \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha}\sum _{m=n}^{\infty}\int _{m^{\alpha}}^{(m+1)^{\alpha}} m^{-\frac{\alpha p}{2}}E|X|^{\frac {p}{2}}I\bigl(|X|>u\bigr)\,du \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha}\sum _{m=n}^{\infty}\int _{m^{\alpha}}^{(m+1)^{\alpha}} m^{-\frac{\alpha p}{2}}E|X|^{\frac {p}{2}}I \bigl(|X|>m^{\alpha}\bigr)\,du \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha}\sum _{m=n}^{\infty}m^{-\frac{\alpha p}{2}+\alpha-1}E|X|^{\frac{p}{2}}I \bigl(|X|>m^{\alpha}\bigr) \\ =&C\sum_{m=1}^{\infty}m^{-\frac{\alpha p}{2}+\alpha-1}E|X|^{\frac {p}{2}}I \bigl(|X|>m^{\alpha}\bigr)\sum_{n=1}^{m} n^{\alpha p-1-\alpha} \\ \leq&C\sum_{m=1}^{\infty}m^{\frac{\alpha p}{2}-1} E|X|^{\frac {p}{2}}I\bigl(|X|>m^{\alpha}\bigr) \\ =&C\sum_{m=1}^{\infty}m^{\frac{\alpha p}{2}-1}\sum _{n=m}^{\infty}E|X|^{\frac{p}{2}}I \bigl(n^{\alpha}< |X|\leq(n+1)^{\alpha}\bigr) \\ \leq&C\sum_{n=1}^{\infty}E|X|^{\frac{p}{2}}I \bigl(n^{\alpha}< |X|\leq (n+1)^{\alpha}\bigr)\sum _{m=1}^{n} m^{\frac{\alpha p}{2}-1} \\ \leq&C\sum_{n=1}^{\infty}n^{\frac{\alpha p}{2}} E|X|^{\frac {p}{2}}I\bigl(n^{\alpha}< |X|\leq(n+1)^{\alpha}\bigr) \\ \leq&C E|X|^{p}< \infty. \end{aligned}$$
(2.7)
$$\begin{aligned} I \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-\frac{q}{2}}E\Biggl(\max _{1\leq j\leq n}\Biggl|\sum_{i=1}^{j} X_{ui}\Biggr|\Biggr)^{\frac {q}{2}}\,du \quad \mbox{by Markov inequality} \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-\frac{p}{2}}\sum _{i=1}^{n} E|X_{ui}|^{\frac{p}{2}}\,du \quad \biggl(q=p \mbox{ and } \frac{p}{2}< \frac{1}{2}\biggr) \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-\frac{p}{2}}\sum _{i=1}^{n} \bigl\{ E|X_{i}|^{\frac{p}{2}}I\bigl(|X_{i}| \leq u\bigr)\bigr\} \,du \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha} \int_{n^{\alpha}}^{\infty}u^{-\frac{p}{2}}E|X|^{\frac{p}{2}}I\bigl(|X| \leq u\bigr)\,du \\ &{} +C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha} \int_{n^{\alpha}}^{\infty}P\bigl(|X|>u\bigr)\,du \quad\mbox{by Lemma}~2.2(\mathrm{ii}) \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha} \int_{n^{\alpha}}^{\infty}u^{-\frac{p}{2}}E|X|^{\frac{p}{2}}I\bigl(|X| \leq u\bigr)\,du \\ &{}+C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha} \int_{n^{\alpha}}^{\infty}u^{-\frac{p}{2}}E|X|^{\frac{p}{2}}I\bigl(|X|>u\bigr)\,du \\ =&I_{1}+I_{2}. \end{aligned}$$
$$\begin{aligned} I_{1} =&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha}\sum_{m=n}^{\infty}\int _{m^{\alpha}}^{(m+1)^{\alpha}}u^{-\frac{p}{2}}E|X|^{\frac{p}{2}}I\bigl(|X| \leq u\bigr)\,du \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha}\sum _{m=n}^{\infty}m^{-\frac{\alpha p}{2}+\alpha-1} E|X|^{\frac{p}{2}}I\bigl(|X|\leq (m+1)^{\alpha}\bigr) \\ =&C\sum_{m=1}^{\infty}m^{-\frac{\alpha p}{2}+\alpha-1} E|X|^{\frac {p}{2}}I\bigl(|X|\leq(m+1)^{\alpha}\bigr)\sum _{n=1}^{m} n^{\alpha p-1-\alpha} \\ \leq&C\sum_{m=1}^{\infty}m^{\frac{\alpha p}{2}-1}E|X|^{\frac {p}{2}}I \bigl(|X|\leq(m+1)^{\alpha}\bigr) \\ \leq&C\sum_{m=1}^{\infty}m^{\frac{\alpha p}{2}-1} \sum_{n\leq m} E|X|^{\frac{p}{2}}I\bigl(n^{\alpha}< |X| \leq(n+1)^{\alpha}\bigr) \\ \leq&C\sum_{m=1}^{\infty}m^{\frac{\alpha p}{2}-1} \sum_{n\leq m}n^{\frac {\alpha p}{2}}P\bigl(n^{\alpha}< |X| \leq(n+1)^{\alpha}\bigr) \\ \leq&C\sum_{m=1}^{\infty}m^{\alpha p}P \bigl(m^{\alpha}< |X|\leq(m+1)^{\alpha}\bigr) \leq CE|X|^{p}< \infty. \end{aligned}$$
(2.8)
In the case \(1\leq p < 2\), let \(Y_{ui}=X_{i}I(|X_{i}|\leq u)-uI(X_{i} < -u)+uI(X_{i}>u)\) for all \(u>0\), and \(Y_{ui}^{\prime}=X_{i}-Y_{ui}\), \(i\geq1\). Then we have Note that \(\{Y_{ui}-EY_{ui}\}\) and \(\{Y_{ui}^{\prime}-EY_{ui}^{\prime}\}\) are sequences of negatively associated random variables.
$$\begin{aligned} &\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}P\Biggl(\max_{1\leq k\leq n}\Biggl| \sum_{i=1}^{k} X_{i}\Biggr|>u\Biggr)\,du \\ &\quad\leq\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}P\Biggl(\max_{1\leq k\leq n}\Biggl| \sum_{i=1}^{k} (Y_{ui}-EY_{ui})\Biggr|> \frac{u}{2}\Biggr)\,du \\ &\qquad{} +\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}P\Biggl(\max_{1\leq k\leq n}\Biggl| \sum_{i=1}^{k}\bigl(Y_{ui}^{\prime}-EY_{ui}^{\prime} \bigr)\Biggr|>\frac {u}{2}\Biggr)\,du \quad\mbox{since }EX_{n}=0 \\ &\quad:=I^{\prime}+J^{\prime}. \end{aligned}$$
(2.9)
For \(J^{\prime}\), take q such that \(1\leq p< q\leq2\). By the fact that \(|Y_{ui}^{\prime}|\leq|X_{i}|I(|X_{i}|>u)\), the Markov inequality, Lemma 2.1, (2.1), Lemma 2.2(iii), (2.3), and the \(C_{r}\)-inequality we have
$$\begin{aligned} J^{\prime} \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q} \sum _{i=1}^{n} E\bigl|Y_{ui}^{\prime}-EY_{ui}^{\prime}\bigr|^{q}\,du \quad\mbox{by}~(2.1) \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}\sum _{i=1}^{n} E\bigl|Y_{ui}^{\prime}\bigr|^{q}\,du \quad\mbox{(by the $C_{r}$-inequality)} \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}\sum _{i=1}^{n} E|X_{i}|^{q}I\bigl(|X_{i}|>u\bigr)\,du \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}E|X|^{q}I\bigl(|X|>u\bigr)\,du \quad\mbox{Lemma}~2.2(\mathrm{iii}) \\ =&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha}\sum _{m=n}^{\infty}\int _{m^{\alpha}}^{(m+1)^{\alpha}} u^{-q}E|X|^{q}I\bigl(|X|>u\bigr)\,du \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha}\sum _{m=n}^{\infty}m^{-\alpha q+\alpha-1}E|X|^{q}I \bigl(|X|>m^{\alpha}\bigr) \\ \leq&C\sum_{m=1}^{\infty}m^{-\alpha q+\alpha-1}E|X|^{q}I \bigl(|X|>m^{\alpha}\bigr)\sum_{n=1}^{m} n^{\alpha p-1-\alpha} \\ =&C\sum_{m=1}^{\infty}m^{-\alpha(q-p)-1} E|X|^{q}I\bigl(|X|>m^{\alpha}\bigr) \\ =&C\sum_{m=1}^{\infty}m^{-\alpha(q-p)-1}\sum _{n=m}^{\infty}E|X|^{q}I \bigl(n^{\alpha}< |X|\leq(n+1)^{\alpha}\bigr) \\ \leq&C\sum_{n=1}^{\infty}E|X|^{q}I \bigl(n^{\alpha}< |X|\leq(n+1)^{\alpha}\bigr)\sum _{m=1}^{n} m^{-\alpha(q-p)-1} \\ \leq&C\sum_{n=1}^{\infty}E|X|^{p}I \bigl(n^{\alpha}< |X|\leq(n+1)^{\alpha}\bigr) \\ \leq&C E|X|^{p}< \infty. \end{aligned}$$
(2.10)
For \(I^{\prime}\), take q such that \(1< p< q\leq2\). By the Markov inequality, (1.2), Lemma 2.1, (2.1), the \(C_{r}\)-inequality, and Lemma 2.2(iv) we have
$$\begin{aligned} I^{\prime} \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}E\Biggl(\max _{1\leq k\leq n}\sum_{i=1}^{k}|Y_{ui}-EY_{ui}|^{q} \Biggr)\,du \quad\mbox{by Markov inequality} \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}\sum _{i=1}^{n} E|Y_{ui}-EY_{ui}|^{q} \,du \quad\mbox{by}~(2.1) \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}\sum _{i=1}^{n} E|Y_{ui}|^{q}\,du \quad \mbox{by the $C_{r}$-inequality} \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}E|X|^{q}I\bigl(|X| \leq u\bigr)\,du \\ &{} +C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}E|X|^{q}I\bigl(|X|>u\bigr)\,du \quad \mbox{by Lemma}~2.2(\mathrm{iv}) \\ =&I_{1}^{\prime}+I_{2}^{\prime}. \end{aligned}$$
According to the calculation of \(J^{\prime}\), we obtain \(I_{2}^{\prime}<\infty\) (see (2.10)). It remains to prove that \(I_{1}^{\prime}<\infty\). By taking q such that \(1\leq p< q\leq2\) we have Hence, from (2.9)-(2.11) the result (2.5) follows in the case \(1\leq p< 2\).
$$\begin{aligned} I_{1}^{\prime} =&C\sum _{n=1}^{\infty}n^{\alpha p-1-\alpha}\sum _{m=n}^{\infty}\int _{m^{\alpha}}^{(m+1)^{\alpha}}u^{-q}E|X|^{q}I\bigl(|X| \leq u\bigr)\,du \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha}\sum _{m=n}^{\infty}m^{-\alpha q+\alpha-1} E|X|^{q}I\bigl(|X|\leq(m+1)^{\alpha}\bigr) \\ =&C\sum_{m=1}^{\infty}m^{-\alpha q+\alpha-1} E|X|^{q}I\bigl(|X|\leq (m+1)^{\alpha}\bigr)\sum _{n=1}^{m} n^{\alpha p-1-\alpha} \\ \leq&C\sum_{m=1}^{\infty}m^{-\alpha q+\alpha p-1}E|X|^{q}I \bigl(|X|\leq (m+1)^{\alpha}\bigr) \\ \leq&C\sum_{m=1}^{\infty}m^{-\alpha q+\alpha p-1}\sum _{n\leq m} E|X|^{q}I\bigl(n^{\alpha}< |X| \leq(n+1)^{\alpha}\bigr) \\ \leq&C\sum_{m=1}^{\infty}m^{\alpha p}P \bigl(m^{\alpha}< |X|\leq(m+1)^{\alpha}\bigr) \\ =&CE|X|^{p}< \infty. \end{aligned}$$
(2.11)
In the case \(p\geq2\), we also obtain \(I^{\prime}\) and \(J^{\prime}\) of (2.9).
For \(I^{\prime}\), by the Markov inequality, the \(C_{r}\)-inequality, and Lemma 2.1 (2.2) we have that, for \(q>2\), We will consider \(I_{3}^{\prime}\) and \(I_{4}^{\prime}\) as follows.
$$\begin{aligned} I^{\prime} \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}E\Biggl\{ \max _{1\leq j\leq n}\Biggl|\sum_{i=1}^{j}(Y_{ui}-EY_{ui})\Biggr| \Biggr\} ^{q} \,du \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}\Biggl\{ \sum _{i=1}^{n}E|Y_{ui}-EY_{ui}|^{q} +\Biggl(\sum_{i=1}^{n}E|Y_{ui}-EY_{ui}|^{2} \Biggr)^{\frac{q}{2}}\Biggr\} \,du \quad\mbox{by}~(2.2) \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}\sum _{i=1}^{n}E|Y_{ui}|^{q} \,du \\ &{} +C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}\Biggl(\sum _{i=1}^{n} EY_{ui}^{2} \Biggr)^{\frac{q}{2}}\,du \quad\mbox{by the $C_{r}$-inequality} \\ =&I_{3}^{\prime}+I_{4}^{\prime}. \end{aligned}$$
(2.12)
Note that \(\alpha>\frac{1}{2}\), \(\alpha p>1\), and \(p\geq2\). Take \(q>\max(p, \frac{\alpha p-1}{\alpha-\frac{1}{2}})\), which implies that \(\alpha p-2-\alpha q+\frac{q}{2}<-1\).
By Lemma 2.2(iv) we have According to the calculation of \(J^{\prime}\), we obtain \(I_{32}^{\prime}<\infty\) (see (2.10)). It remains to prove \(I_{31}^{\prime}\): which yields \(I_{3}^{\prime}<\infty\).
$$\begin{aligned} I_{3}^{\prime} \leq&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}E|X|^{q}I\bigl(|X| \leq u\bigr)\,du \\ &{}+C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}E|X|^{q}I\bigl(|X|> u\bigr)\,du \\ \leq&I_{31}^{\prime}+I_{32}^{\prime}. \end{aligned}$$
$$\begin{aligned} I_{31}^{\prime} \leq&C\sum _{n=1}^{\infty}n^{\alpha p-1-\alpha}\sum _{m=n}^{\infty}\int_{m^{\alpha}}^{(m+1)^{\alpha}} u^{-q}E|X|^{q}I\bigl(|X| \leq u\bigr)\,du \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha}\sum _{m=n}^{\infty}m^{\alpha-1-\alpha q}E|X|^{q}I \bigl(|X|\leq(m+1)^{\alpha}\bigr) \\ =&C\sum_{m=1}^{\infty}m^{\alpha-1-\alpha q}E|X|^{q}I \bigl(|X|\leq(m+1)^{\alpha}\bigr)\sum_{m=1}^{n} m^{\alpha p-1-\alpha} \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-1-\alpha q}E|X|^{q}I \bigl(|X|\leq (m+1)^{\alpha}\bigr) \\ \leq&C\sum_{m=1}^{\infty}m^{\alpha p-1-\alpha q}E|X|^{q}I \bigl(m^{\alpha}< |X|\leq(m+1)^{\alpha}\bigr) \\ &{} +C\sum_{m=1}^{\infty}m^{\alpha p-1-\alpha q}E|X|^{q}I \bigl(|X|\leq m^{\alpha}\bigr) \\ \leq&C\sum_{m=1}^{\infty}m^{-1} E|X|^{p}I\bigl(m^{\alpha}< |X|\leq(m+1)^{\alpha}\bigr) \\ &{} +C\sum_{m=1}^{\infty}m^{-\alpha(q-p)-1}\sum _{j=1}^{m} j^{\alpha q}P \bigl((j-1)^{\alpha}< |X|\leq j^{\alpha}\bigr) \\ \leq&C\sum_{m=1}^{\infty}E|X|^{p}I \bigl(m^{\alpha}< |X|\leq(m+1)^{\alpha}\bigr) \\ &{} +C\sum_{j=1}^{\infty}j^{\alpha q}P \bigl((j-1)^{\alpha}< |X|\leq j^{\alpha}\bigr)\sum _{m=j}^{\infty}m^{-\alpha(q-p)-1} \\ \leq&CE|X|^{p}+C\sum_{j=1}^{\infty}j^{\alpha p}P\bigl((j-1)^{\alpha}< |X|\leq j^{\alpha}\bigr) \\ \leq&CE|X|^{p}< \infty, \end{aligned}$$
(2.13)
By the fact that \(EX^{2}<\infty\) and \(E|X|^{p}<\infty\) we have for \(p\geq2\): Thus, the proof of Lemma 2.4 is complete. □
$$\begin{aligned} I_{4}^{\prime} \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}\Biggl(\sum _{i=1}^{n}EX_{i}^{2} \Biggr)^{\frac{q}{2}}\,du \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}u^{-q}\bigl(n EX^{2}\bigr)^{\frac{q}{2}}\,du \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha+\frac{q}{2}} \int_{n^{\alpha}}^{\infty}u^{-q}\,du \\ \leq&C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha q+\frac{q}{2}}< \infty. \end{aligned}$$
3 Main results
Theorem 3.1
Let
\(\alpha p> 1\), \(p>0\), and
\(\alpha >\frac{1}{2}\). Let
\(\{X_{n}, n\geq1\}\)
be a sequence of negatively associated random variables with
\(EX_{n}=0\)
for all
\(n\geq1\), and
X
be a random variable satisfying conditions (1.2) and
\(E|X|^{p}<\infty\). Then, for all
\(\epsilon>0\),
where
\(a^{+}=\max\{a, 0\}\).
$$ \sum_{n=1}^{\infty}n^{\alpha p-2-\alpha}E\Biggl( \max_{1\leq k\leq n}\Biggl|\sum_{i=1}^{k} X_{i}\Biggr|-\epsilon n^{\alpha}\Biggr)^{+}< \infty, $$
(3.1)
Proof
To prove (3.1), we use Lemmas 2.3 and 2.4 as follows: By Lemma 2.3 the first term of (3.2) is finite, and by Lemma 2.4 the second term of (3.2) is finite. Hence, the proof of the theorem is complete. □
$$\begin{aligned} &\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha}E\Biggl(\max_{1\leq k\leq n}\Biggl|\sum _{i=1}^{k} X_{i}\Biggr|-\epsilon n^{\alpha}\Biggr)^{+} \\ &\quad=\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{0}^{\infty}P\Biggl(\Biggl(\max _{1\leq k\leq n}\Biggl|\sum_{i=1}^{k} X_{i}\Biggr|-\epsilon n^{\alpha}\Biggr)^{+}>u\Biggr)\,du \\ &\quad=\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{0}^{\infty}P\Biggl(\max_{1\leq k\leq n}\Biggl| \sum_{i=1}^{k} X_{i}\Biggr|-\epsilon n^{\alpha}>u\Biggr)\,du \\ &\quad=\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{0}^{n^{\alpha}} P\Biggl(\max_{1\leq k\leq n}\Biggl| \sum_{i=1}^{k} X_{i}\Biggr|-\epsilon n^{\alpha}>u\Biggr)\,du \\ &\qquad{} +\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}P\Biggl(\max_{1\leq k\leq n}\Biggl| \sum_{i=1}^{k} X_{i}\Biggr|-\epsilon n^{\alpha}>u\Biggr)\,du \\ &\quad\leq\sum_{n=1}^{\infty}n^{\alpha p-2} P\Biggl(\max_{1\leq k\leq n}\Biggl|\sum_{i=1}^{k} X_{i}\Biggr|>\epsilon n^{\alpha}\Biggr) \\ &\qquad{} +\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{n^{\alpha}}^{\infty}P\Biggl(\max_{1\leq k\leq n}\Biggl| \sum_{i=1}^{k} X_{i}\Biggr|>u\Biggr)\,du. \end{aligned}$$
(3.2)
Theorem 3.2
Let \(\alpha p> 1\), \(p>0\), and \(\alpha >\frac{1}{2}\). Let \(\{X_{n}, n\geq1\}\) be a sequence of negatively associated random variables with \(EX_{n}=0\) for all \(n\geq1\), and X be a random variable possibly defined on a different space satisfying the condition (1.2). Then (3.1) implies (2.4).
Proof
It is easy to see that Hence, by (3.3), (3.1) implies (2.4). The proof of the theorem is complete. □
$$\begin{aligned} &\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha}E\Biggl(\max_{1\leq k\leq n}\Biggl|\sum _{i=1}^{k} X_{i}\Biggr|-\epsilon n^{\alpha}\Biggr)^{+} \\ &\quad\geq\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{0}^{\infty}P\Biggl(\max_{1\leq k\leq n}\Biggl| \sum_{i=1}^{k} X_{i}\Biggr|-\epsilon n^{\alpha}>u\Biggr)\,du \\ &\quad\geq\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{0}^{\epsilon n^{\alpha}} P\Biggl(\max_{1\leq k\leq n}\Biggl| \sum_{i=1}^{k} X_{i}\Biggr|>\epsilon n^{\alpha}+u\Biggr)\,du \\ &\quad\geq\epsilon\sum_{n=1}^{\infty}n^{\alpha p-2} P\Biggl(\max_{1\leq k\leq n}\Biggl|\sum _{i=1}^{k} X_{i}\Biggr|>2\epsilon n^{\alpha}\Biggr). \end{aligned}$$
(3.3)
Remark
According to Ko [14], the results on the complete moment convergence for negatively associated random variables was established only in the case \(p>1\). In this paper, it is obtained in the case \(p>0\).
Theorem 3.3
Let
\(\alpha p>1\), \(p>0\), and
\(\alpha>\frac {1}{2}\). Let
\(\{X_{n}, n\geq1\}\)
be a sequence of negatively associated random variables with
\(EX_{n}=0\)
for all
\(n\geq1\), and
X
be a random variable satisfying condition (1.2) and
\(E|X|^{p}<\infty\). Then, for all
\(\epsilon>0\),
$$ \sum_{n=1}^{\infty}n^{\alpha p-2}E\Biggl\{ \sup_{k\geq n}\Biggl|k^{-\alpha}\sum_{j=1}^{k} X_{j}\Biggr|-\epsilon\Biggr\} ^{+}< \infty. $$
(3.4)
Proof
By (3.1) we have where \(\epsilon^{\prime}=\epsilon2^{-\alpha}\). Hence, the proof of (3.4) is completed. □
$$\begin{aligned} &\sum_{n=1}^{\infty}n^{\alpha p-2} E\Biggl\{ \sup_{k\geq n}\Biggl|k^{-\alpha}\sum_{i=1}^{k}X_{i}\Biggr|- \epsilon\Biggr\} ^{+} \\ &\quad=\sum_{n=1}^{\infty}n^{\alpha p-2} \int_{0}^{\infty}P\Biggl(\sup_{k\geq n}\Biggl|k^{-\alpha} \sum_{i=1}^{k}X_{i}\Biggr|>\epsilon+u \Biggr)\,du \\ &\quad=\sum_{j=1}^{\infty}\sum _{n=2^{j-1}}^{2^{j}-1}n^{\alpha p-2} \int _{0}^{\infty}P\Biggl(\sup_{k\geq n}\Biggl|k^{-\alpha} \sum_{i=1}^{k}X_{i}\Biggr|>\epsilon+u \Biggr)\,du \\ &\quad\leq C\sum_{j=1}^{\infty}\int_{0}^{\infty}P\Biggl(\sup_{k\geq2^{j-1}}\Biggl|k^{-\alpha } \sum_{i=1}^{k}X_{i}\Biggr|>\epsilon+u \Biggr)\,du \sum_{n=2^{j-1}}^{2^{j}-1}n^{\alpha p-2} \\ &\quad\leq C\sum_{j=1}^{\infty}2^{j(\alpha p-1)} \int_{0}^{\infty}P\Biggl(\sup_{k\geq 2^{j-1}}\Biggl|k^{-\alpha} \sum_{i=1}^{k}X_{i}\Biggr|>\epsilon+u \Biggr)\,du \\ &\quad\leq C\sum_{j=1}^{\infty}2^{j(\alpha p-1)} \sum_{m=j}^{\infty}\int _{0}^{\infty}P\Biggl(\max_{2^{m}-1\leq k\leq2^{m}}\Biggl|k^{-\alpha} \sum_{i=1}^{k}X_{i}\Biggr|>\epsilon+u \Biggr)\,du \\ &\quad\leq C\sum_{m=1}^{\infty}\int_{0}^{\infty}P\Biggl(\max_{2^{m}-1\leq k\geq 2^{m}}\Biggl|k^{-\alpha} \sum_{i=1}^{k}X_{i}\Biggr|>\epsilon+u \Biggr)\,du \sum_{j=1}^{m} 2^{j(\alpha p-1)} \\ &\quad\leq C\sum_{m=1}^{\infty}2^{m(\alpha p-1)} \int_{0}^{\infty}P\Biggl\{ \max_{2^{m}-1\leq k\leq2^{m}}\Biggl| \sum_{i=1}^{k}X_{i}\Biggr|>( \epsilon+u)2^{(m-1)\alpha}\Biggr\} \,du \\ &\qquad \bigl(\mbox{letting} y=2^{(m-1)\alpha}u\bigr) \\ &\quad\leq C\sum_{m=1}^{\infty}2^{m(\alpha p-1-\alpha)} \int_{0}^{\infty}P\Biggl(\max_{1\leq k\leq2^{m}}\Biggl| \sum_{i=1}^{k}X_{i}\Biggr|> \epsilon2^{(m-1)\alpha}+y\Biggr)\,dy \\ &\quad\leq C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha} \int_{0}^{\infty}P\Biggl(\max_{1\leq k\leq n}\Biggl| \sum_{i=1}^{k}X_{i}\Biggr|>\epsilon n^{\alpha}2^{-\alpha}+y\Biggr)\,dy \\ &\quad=C\sum_{n=1}^{\infty}n^{\alpha p-2-\alpha}E \Biggl(\max_{1\leq k\leq n}\Biggl|\sum_{i=1}^{k}X_{i}\Biggr|- \epsilon^{\prime} n^{\alpha}\Biggr)^{+}< \infty, \end{aligned}$$
Theorem 3.4
Let
\(\alpha>\frac{1}{2}\), \(p>0\), and
\(\alpha p>1\). Let
\(\{X_{n}, n\geq1\}\)
be a sequence of negatively associated random variables with
\(EX_{n}=0\)
for all
\(n\geq1\), and
X
be a random variable satisfying condition (1.2) with
\(E|X|^{p}<\infty\). Then
for all
\(\epsilon>0\).
$$\sum_{n=1}^{\infty}n^{\alpha p-2}P\biggl( \sup_{j\geq n}\biggl|\frac{S_{j}}{j^{\alpha}}\biggr|\geq\epsilon\biggr)< \infty $$
Proof
Inspired by the proof of Theorem 12.1 of Gut [15], we can prove it and omit the proof. □
Declarations
Acknowledgements
The author is exceptionally grateful to the referee for offering helpful comments and suggestions. This paper was supported by Wonkwang University in 2016.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
(1)
Division of Mathematics and Informational Statistics, Nanoscale Sciences and Technology Institute, Wonkwang University, Jeonbuk, Korea
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